Page 34 - 4749

x
N
u   x   x dx C . (2.22)
o EA
Integration constants in areas are determined from the boundary
conditions (conditions of rod fixing) and the conditions of
connecting areas between each other (conditions of continuity of
displacement functions in the section in the transition from one
area to another).
In the equation (2.22) we find the constant of integration C
with the boundary conditions: x    0u  u . So C  u –
0
o o
displacement of the left end of the rod, then
x N
u   x  u  x dx . (2.23)
o 
o EA
If in the area of a rod   N A const , then
x x
N
u   x  u  x x, (2.24)
o
EA
that is, in this area the displacement is changing linearly and the
deformation   const .
x
If the rod is loaded by longitudinal loads so that there are
several areas, so, on every area we should:
- at constant stress   const , use the equation (2.24);
x
- at variable stress   const , use the equation (2.23).
x

2.6 Consideration of rod own weight

Consider the rod of stable cross-sectional area, fixed by the top
end and loaded only by its own weight (fig.2.8 a).
A design scheme of the rod can be represented as a weightless
rod that loaded by evenly distributed longitudinal load with the
intensity q   gA  A (fig.2.8, b), where – unit weight of a
material rod.
34